## ABSTRACT

The current study deals with a composite broad-crested weir which is specially designed with the unique idea for a ‘constant discharge coefficient (*C _{d}*) of 0.6’. It is investigated experimentally and numerically. The available designs of the weir are unable to give constant

*C*over a wide range of discharge as

_{d}*‘C*’ itself is relative to the head over the weir crest. Therefore an attempt is made to restrict

_{d}*C*value to 0.6 irrespective of the variable head on the weir crest. This is achieved by adjusting the widths of the weir. With the novel objective,

_{d}*C*is frozen to constant value and instead of it,

_{d}*‘b*’ is allowed to vary. The weir so designed is capable of producing constant

*C*over a wide range of discharge and hence will be helpful from the viewpoint of field applications. Under existing laboratory conditions, the research reports for emissions varying from 20 to 100% of the design discharge. The numerical performance of the CBC weir through FLOW 3D is experimentally validated to examine the crest width effect and head over weir crest. In the experiments,

_{d}*C*is found to vary proportionally with discharge from 0.518 to 0.648. The

_{d}*R-*value is 0.999, with a mean error in discharge measurement being much less.

## HIGHLIGHTS

The available designs of the weir are unable to give constant

*C*over a wide range of discharge as ‘_{d}*C*’ itself is relative to head over weir crest._{d}Thus, an attempt was made to restrict the

*C*value (0.6) irrespective of the variable head on weir crest._{d}This is achieved by controlling head over weir by adjusting the widths of weir.

With the novel objective,

*C*is frozen to a constant value and instead,_{d}*‘b*’ is allowed to vary.

## NOMENCLATURE

- CBC
composite broad crested

*C*_{d}coefficient of discharge

- Fr
Froude number

- Fr
_{1} supercritical or initial Froude number

*g*acceleration due to gravity

*H*velocity head on upstream of the weir

*h/b*head over weir crest to weir width ratio

*H*_{o}total energy head at the

*u/s*end*L*length of the weir

- LOC
level of confidence

- PLA
polylactic acid

- PVC
polyvinyl chloride

*Q*weir discharge

*q*discharge per unit width of weir

*Q*_{max}maximum discharge

*Q*_{min}minimum discharge

- VOF
volume of fluid

## INTRODUCTION

Weirs with broad crests are fundamental structures for measuring flow in various widths of channels. It is a flat-crested structure having a greater crest length in relation to flow thickness, which results in a hydrostatic pressure dispersion. Weirs with sharp crests and broad crests have always gained the interest of researchers. The open medium flowing gauging has long been investigated by researchers like Ackers (1978) and RangaRaju (2001), who discovered flowing resources of limited crest length weirs. Emissions correlation for flowing values with cross-section structures like rectangular, triangular, trapezoidal, and truncated triangular are focussed upon and the needed experiential emission formulae for such weirs have been suggested in the past few years (Boiten & Pitlo 1982; Hager & Schwalt 1994; Gogus *et al*. 2016). Kindsvater & Carter (1959) produced an equation to generate a trusted, reliable technique for creating every rectangular model (subdued, partly condensed, and wholly condensed weirs). Kulin & Compton (1975) initiated a grading method for partly condensed 90° and completely condensed V notch weirs between 25° and 100° that can also be utilized on totally subdued, partly condensed, and wholly condensed rectangular weirs. It was proposed to do head–discharge measurements for broad weirs and lengthy canyons with various cross patterns and this is evident through studies conducted by Bos (1989). When consistency and exterior tension are not accounted for, the emission coefficient would be insistent on the weir head over the crest to length ratio, *h/b* (Horton 1907). Harrison (1967) conducted an assertion for the coefficient of emission for an aerodynamic wide-crested weir depending on the theory of critical flow which accounted for frontier coating formation. Lab estimates as directed by Gogus *et al*. (2006) observed that the universal coefficient of discharge for a composite wide-crested weir is lower than that of a plain wide-crested weir with the same level of crest height and lengths. Studies were conducted on emission links for rectangular wide-crested weirs with an aim of testing the impact of subsidiary weir crest breadth and step height of wide-crested weirs with rectangular compound samplings on the measures of emission coefficient and proposition velocity coefficient (Azimi & Rajaratnam 2009; Salmasi *et al*. 2011; BijanKhan *et al.* 2014). The use of mathematical software and soft computing strategies have been lately used by researchers in the area of open-channel hydraulics, especially for compound weirs. Hinge *et al*. (2010, 2011) signified calibrating a compound weir with a tiny stretch in the midst of a hydraulic jump. During challenging flow circumstances, the upstream flow levels were calculated for the provided flow rates using the commercial program FLUENT 6.3.26. The outcomes were deemed to be excellent. Predicting discharge in composite channels using artificial intelligence approaches such as linear genetic programming (LGP) and M5 model trees to substantiate the empirical and traditional methods has piqued the interest of academics all around the world. Mathematical models were analyzed and they indicated good coefficient of emission predictions with errors less than 3.8% and enhanced outputs. Hager & Schwalt (1994) have made similar efforts. Previous research has linked the *C _{d}* to the weir head. Savage & Johnson (2001) determined the emission coefficient for the Ogee spillway by combining 2D and 3D real-life models with FLOW 3D simulations. The 2D model yielded acceptable outcomes. Using the STAR-CD program, Khan

*et al*. (2006) generated velocity vs. dimension plots on a contact tank. Top, center, and bottom depth graphs were created. In Flow 3D modeling, the turbulent flow model has five schemes: large eddy simulation (LES), renormalization group (RNG), the two equations (

*k–ɛ*), one equation turbulent energy (

*k*), and Prandtl's mixing length theory. The RNG turbulence model was used to perform CFD by Flow 3D and simulations resulted in the discharge values based on upstream flow depth. The velocity field was accurately represented using the CFD STAR-CD model. Rady (2011) utilized the FLOW-3D application to create a rectangular pointed-crested weir with

*C*as a utility of weir head-to-weir height ratio. The determined

_{d}*C*value was reported to be within 3% inaccuracy. The benefits of implementing ‘FLOW-3D’ were also emphasized. These uses, however, were examined for rectangle sharp-crested weirs. The head was displayed utilizing the rate of flow vs. FLUENT, and both the observational and computational accumulation were in good agreement. Bilhan

_{d}*et al*. (2018) carried out the framework of

*C*estimates for a spheric labyrinth weir using and excluding a nappe cutter utilizing FLOW-3D. The outcomes of the experiment and CFD results were nearly the same, with 4% average variance barely. FLOW-3D's pertinence to wide-crested permeable weirs is tested by Safarzadeh & Mohajeri (2018). The coefficient of drag was computed using the experimental head–discharge curve, which helped in identifying the connection between the coefficient of resistance and the coefficient of impermeability. Kulkarni & Hinge (2017, 2020) have piloted substantial empirical tasks and have come to the conclusion that a wide-crested weir sample can be used as emission calculating instrument whilst holding

_{d}*C*as invariant. Kulkarni & Hinge (2021a, 2021b) have lately come up with the utilization of compound weir for precise estimations with the help of addible manufacturing for CBC weir geometry having different base width variations leading to fairly simulated discharge values. This study also led to the conclusion that FLOW-3D is an efficient tool for effective discharge predictions using numerical analysis.

_{d}Soft computing and machine learning techniques have been increasingly used in the area of pipe and open-channel hydraulics. The use of a rectangular compound side weir for calculating a wide range of discharges was conceived by Zahiri *et al*. (2013). The discharge values that were sought, ranged from 1.7 to 16.3 L per second (lps) in an inclined flume with a length of 9 m, a width of 30 cm, and a height of 50 cm. Froude and Reynolds' numbers of flow varied between 0.42 and 0.96, and between 2,500 and 19,000 correspondingly. Zahiri *et al*. (2014) discovered the M5 tree decision model to be applied to the hydraulic model study of unwavering uniform flows in conjugated open transmissions using LGP. Large computational efforts based on the mathematical solution of differential equations for compound channels due to different geometric and hydraulic conditions were addressed. To simplify the analysis, parameters such as depth ratio, coherence, and estimated flow discharges to bank full discharge ratio were employed. Mathematical models were also analyzed, and they demonstrated better approximation of emission coefficients with a fallacy lower than 3.8%, with superior outcomes provided by numerical and soft computing tools.

Recently, Roushangar *et al*. (2021) used a metaheuristic technique called Grey Wolf Optimization to reduce the cross-section of compound trapezoidal channels, allowing for an in-depth examination of flow characteristics. A novel design in triangular broad-crested weir theory has been proposed with rigorous experimentation work by Achour & Amara (2022). A detailed study of the semi-modular triangular broad-crested weir is presented which can be used as a self-cleansing device as well. Discharge relations established and resulting discharge coefficient are validated with a high coefficient of determination indicating very minor correction required for the theoretical stage-discharge formula, requiring no further addition of calibration parameter. Azamathulla *et al*. (2016) stated that assisted vector machine methods are equally helpful for anticipating side weir discharge coefficient. Optimization methods like PSO were analyzed by Parsaie *et al*. (2018) to decide *C _{d}* of the cylindric weir passageway. Achour & Amara (2022) came to a conclusion that the ratio

*h/B*(weir head-to-width ratio) accounts for 23.5% as a standard consequence in the measuring of the emission coefficient (

*C*) and therefore of the flow rate

_{d}*Q*. Corresponding attempts directed on rectangular and triangular mediums utilizing wide-crested and slender-crested sill elucidate upon hydraulic jump attributes in surface flow hydraulics by Achour

*et al*. (2022a, 2022b, 2022c). The flow over a rectangular weir positioned at the bottom of a canal was numerically simulated using the FLUENT software (version 6.3.1) by Mohammadpour

*et al*. (2013). The effects of various turbulence models on flow field prediction were examined in the study. The numerical modeling of this study was predicated on the presumption that the flow inside the porous media was laminar, ignoring the inertia effect generated by flow velocity. The flow was simulated using the standard Darcy's law. Data-driven techniques for weir study have recently been the subject of study for many researchers. Fuladipanah

*et al*. (2023) predicted the scour depth downstream of the flip-bucket spillway. A comprehensive evaluation of data-driven models (Support Vector Machine (SVM), Gene Expression Programing (GEP), Multi-Layer Perceptron (MLP), and Multivariate Adaptive Regression Splines (MARS)) vs. five experimental models for predicting scour depth downstream of flip-bucket weir was explained proving these models having upper hand over experimental notions.

It is confirmed that unnecessary fluctuations in *C _{d}* may cause design irrelativity resulting in hefty and costly structures. It also causes losses due to transitioning, contraction, and friction impacts. To make the most use of the water available, a hydraulic structure that is both hydraulically and financially efficient must be designed, built, and evaluated for discharge assessment. An empirically constructed, experimentally established, and mathematically verified relation of head and discharge needs to be devised to calculate the rate of discharge. The use of 3D modeling approaches for discharge evaluation is recommended according to literature based on computational studies on open-channel flows. The FLOW-3D approach has been shown to be promising, however, it must be thoroughly examined before making any major applications of crested weirs. The present work is about the creation of an innovative broad-crested weir for emission measuring in order to achieve this goal. The novel technique of holding an unvarying emission coefficient is utilized to properly measure a broad array of emissions no matter the weir head, where the coefficient of discharge is one of the input design parameters. The experiments were taken up with the notion of illustrating the creative concept of sustaining a consistent estimate of

*C*no matter the head above weir presented in this study. The current study focuses on the mathematical model formulation, fabrication, and performance for the compound broad crest weir. The validation and comparison of outcomes from experiments with theoretical input via CFD simulation and statistical methods are highlighted. According to the discharge predictions offered by the literature study, the emission coefficient (

_{d}*C*) differs optimally from the

_{d}*H/L*ratio. It signifies that the head over the weir is proportionate. The authors have published the findings of their investigations in order to suggest the innovative notion of holding a constant

*C*estimate no matter the position of the weir head. The aim of this study is to create, construct and examine a novel composite broad-crested (CBC) weir having a combination of rectangular and parabolic weirs, with discharge coefficient as the primary input design parameter. The proposed CBC weir's head discharge relation is investigated experimentally, and its performance is mathematically validated. The CBC weir thus demonstrated can be used to quantify discharge in open-channel flow fields.

_{d}### Comparison of broad-crested weir with other weir types

Broad-crested weirs are robust structures that are usually produced of reinforced concrete and span the entire width of the channel. They are deployed to assess the discharge of open channels and are far superior to sharp-crested weirs for this purpose. The broad-crested weir also has the advantage of operating efficiently with higher downstream water levels than a sharp-crested weir, which is due to the fact that it is a critical depth metre. Broad-crested weirs are structurally robust than sharp-crested weirs and are especially beneficial in situations where sharp-crested weirs are subject to maintenance difficulties. United States Bureau of Reclamation (USBR) states that broad-crested weirs are particularly moulded weirs that can be designed to accommodate more complex channel cross-sections compared to other weirs types. The shape of the control section can be chosen by taking the range of discharge and head variations into account. Some varieties of broad-crested weirs can also transport sediment and pieces of debris more easily than sharp-crested weirs, particularly those with sloped upstream transitions or round noses at upstream parts. Submergence has little influence on the performance of broad-crested weirs by up to 90% with sloped downstream transitions and up to 80% with vertical downstream drops. The following benefits outweigh the disadvantages of the sharp-crested weir:

Structural stability up to 2–3 m head may easily be accomplished.

Concrete may be used as material with steel upstream corners for abrasion protection.

Reduced tailwater submergence effect.

No aeration must be provided.

As a shortcoming, this overflow structure has received less attention, its coefficient of discharge is approximately 30% lower, and more material is required. However, whereas the sharp-crested weir is used for heads up to 0.6 m, with an upper limit of 1 m, and ogee-overflow structures are often too expensive for overflow heads of less than 2–3 m, the broad-crested weir may cover the gap. In the broad-crested category, compound weir is a special weir consisting of two stages, primarily a rectangular shape, triangular shape, and trapezoidal shape (Cipoletti weirs). The weir can be constructed using a combination of different shapes, sizes and a variety of stages. Compound weir is used in situations where flow rates change widely. Practically, a triangular weir might easily handle the normal range of discharges at a structure, but occasionally, much higher flows require a rectangular weir. Table 1 provides a summary of the empirical formulae offered by investigators for predicting discharge for various types of weirs published in the literature.

Sr. no. . | Weir type . | Equation to measure discharge . |
---|---|---|

1 | Sharp crested | |

2 | Broad crested | |

3 | Ogee shaped | |

4 | Contracted | |

5 | Suppressed | |

6 | Rectangular | |

7 | Triangular | |

8 | Trapezoidal |

Sr. no. . | Weir type . | Equation to measure discharge . |
---|---|---|

1 | Sharp crested | |

2 | Broad crested | |

3 | Ogee shaped | |

4 | Contracted | |

5 | Suppressed | |

6 | Rectangular | |

7 | Triangular | |

8 | Trapezoidal |

*Source:*Kulkarni & Hinge 2021a, 2021b

Literature study reveals that the equations for measuring discharge with different types of weirs are expressed as a function of geometrical parameters of the concerned weir shape and cross-section. The prime governing parameters for calculating discharge are weir width, height of weir crest and upstream water head over weir crest. As per literature, some researchers have used ‘*L*’, *‘b*’ or *‘b*_{w}’ to denote the criteria of width of weir. Some even report ‘L’ to be the length of weir parallel to the direction of flow. In the present study, *‘b*’ is referred to be the weir width of intermediate steps of the proposed CBC weir which is measured perpendicular to the direction of flow. *‘B*’ denotes the entire base width of the compound weir corresponding to the width of the channel.

## DESIGN PHILOSOPHY

A single specific discharge can be handled by a single rectangular or triangular weir of a specific height. In the event that a weir is intended for low discharge, it may fail hydraulically due to the effect of drowning. It is possible that a weir built for high discharge will not function properly if it is exposed to lower discharge. It indicates that under a constant head, each discharge requires a specific crest head that is proportionate to the head of discharge. A narrow weir, on the other hand, results in a drop in the height of the vertices. This in turn results in a specific weir width that is inversely proportional to the flow rate for any certain discharge. This means that maintaining the width of a vertex does not maintain its height, and maintaining the height of a vertex does not maintain its width. In order to meet the goal of the study, the shape of the one-sided weir should be built in such a way that it produces the height or breadth effect of all of the individual weirs together. As a result, it is assumed that a rectangular step weir will fulfill this function. Since weirs with broad crest have a greater modular limit (0.85), they are less vulnerable to the effects of submergence, as stated in the USBR Water Measurement Manual. Hence, the design of a broad-crested weir with many rectangular steps is the first step in the research.

### Design governing parameters

Upstream head =

*H*=*y*_{2}as shown in Figure 1Channel width =

*B*Base width of intermediate steps =

*b*Maximum discharge =

*Q*_{max}Minimum discharge =

*Q*_{min}Discharge Coefficient =

*C*= 0.6_{d}Height of weir crest at start =

*y*′Head above weir crest =

*h*

The head on upstream ‘*H*’ is the head of water on the upper side of the sluice gate or it is the height of spillway plus half of the head over spillway crest above stilling basin level for minimum discharge in the range. The head over spillway can be neglected if it is smaller in magnitude because some frictional head loss may occur on spillway surface which may compensate for it. The upstream head comprises head over weir including the velocity head. The head above weir crest at the respective step consists of the step height from the base of the weir (*y*_{2}) minus the starting height of weir crest (*y*′).

### Design philosophy of stepped weir section

Stepped weirs should be designed for specific entries of different discharges and specific input of coefficient of discharge. If free flow exists above the weir, water level at tail end will not affect the flow upstream of the weir. This can be achieved by constructing a rectangular step weir so that each step meets the following conditions:

Condition 1: The perpendicular spacing between the channel's bottommost and any particular step top, upstream of a weir, is equal to the post-jump depth (*y*_{2}) for that particular discharge.

Condition 2: The widths of individual steps should be such that the cumulative discharge passing through the weir, with the water level up to any particular step top, is equal to that particular discharge mentioned in condition 1.

Assumptions made in the design of the CBC weir:

1. Channel is rectangular with horizontal slope.

2. Flow is steady and head on upstream of weir (

*h*) is constant.3. Discharge conditions are varying (variation from 20 to 100% of design discharge).

4. Coefficient of discharge (

*C*) is constant._{d}

### Formulation of mathematical technique

The proposed stepped rectangular (combined) weir ought to take into account a wide range of discharges, from the maximum flow (*Q*_{max} = 100%) to minimal flow equalized to 20% of maximum emission (*Q*_{min}). ‘*N*’ intermediate emissions are considered spanning *Q*_{min} and *Q*_{max} with an interval of (*Q*_{max}–*Q*_{min})/(*N* + 1), leading to (*N* + 2) discharges corresponding to (*N* + 2) steps in a rectangular stepped weir. A combined weir is contemplated to be comprised of various rectangular weirs. The design methodology of the present CBC weir is adopted from the studies included in the doctoral thesis submitted by Hinge (2013) where a computer coding was developed to establish the composite weir geometry consisting of multiple steps, corresponding to a specific discharge coefficient value as design input.

*y*

_{2}is calculated from Belanger momentum equation (Subramanya 2009). From Figure 1, the width of the first step is given by:where,

*y*′ = (

*y*

_{2})

_{1}/4 is designed for

*Q*

_{1}(i.e.,

*Q*

_{min}). (

*y*

_{2})

_{1}is the depth after jump, corresponding to

*Q*

_{1}, given by Belanger momentum equation as stated in the following.where

*Q*, the width of the corresponding step

_{n}*b*can be calculated as follows:where

_{n}*a*represents incremental width at

_{n}*n*th stepand

The free-flow rectangular weir formulas are used in the mathematical procedure. By substituting the composite weir geometry with a smooth curve, the methodology was devised to make the composite weir geometry practicable. All of the step's midpoints are connected with a smooth curve, giving them a parabolic shape until the tenth step's tip. Because there are no *C _{d}* recommendations for stepped weirs,

*C*for free-flow stepped weirs must be developed. In majority cases, the crest of a stepped weir is submerged, necessitating the development of a modified coefficient of outflow that is appropriate to submerged situations.

_{d}_{1}in this model is measured by the technique recommended by Peterka (1984).

*C*is measured with the help of the method stated by Subramanya (2009). While calculating

_{d}*y*

_{2}for every discharge, it is important to compute the relevant prejump depth y1 and consequently the related prejump velocity v1. v1 can be computed using Peterka's approach in USBR Monograph 25 (2001), (www.usbr.gov). This considers the head variation proportional to discharge. For implementation, data of an existing horizontal flume is considered as follows:

*B*= 0.2 m,

*H*= 0.28 m,

*Q*

_{max}= 0.010 cumec,

*Q*

_{min}= 0.002 cumec. Figure 2 demonstrates a mathematically accurate and workable stepped weir layout. As an additive performance,

*C*estimates are measured by the method as stated by Swamee (1989). However, it led to a stepped weir with a zigzag shape. After rigorous testing using various input data and mathematical models, it was shown that even the smallest modification in H and

_{d}*C*could lead to an impractical stepped weir design.

_{d}A review of the literature reveals few efforts on composite wide-crested weirs. For 0.1 ≤ *h/b _{w}* ≤ 0.7, Gogus

*et al*. (2006) investigated several composite broad-crested weir geometries. Gogus

*et al*. added that the

*C*levels are almost constant within a small range for

_{d}*h/b*≥ 0.5. It is observed that the

_{w}*h/b*value for the intermediate weir of step width

_{w}*b*

_{1}contributes over 80% of discharge values for any

*y*

_{2}in its upstream ranges. As a result, with a reasonable estimate, both

*H*and

*C*are considered to be constant throughout the design. It is discovered that for Fr

_{d}_{1}> 4.5, the analytical technique yields a feasible and mathematically valid geometry of a stepped weir. The composite weir is designed with the following elements in mind: ‘

*H*= 0.28 m’,

*‘B*= 0.2 m’, ‘

*Q*

_{max}= 0.01 m

^{3}/s’, ‘

*Q*

_{min}= 0.002 m

^{3}/s’, ‘

*y*’ = 0.017 m’. Rectangular weirs are described in depth by Subramanya (2009). Weirs are categorized into sharp crest, broad crest, long crest, and narrow crest based on their ‘h/b’ value. For the existing weir design and laboratory flume features, the balance between best weir geometry and low manufacturing estimate was acquired at

*C*of 0.6 while constructing the CBC weir of multiple rectangular steps. Table 2 gives the geometry of the designed CBC weir and other parameters related to it. The weir's behavior is examined for a discharge coefficient of 0.6 at boundary conditions.

_{d}Sr.no . | Q (m^{3}/s)
. | y_{1} (m)
. | y_{2} (m)
. | H (m)
. | Fr1 . | b (m)
. |
---|---|---|---|---|---|---|

1 | 0.0020 | 0.004428 | 0.065680 | 0.048680 | 10.8372 | 0.105097 |

2 | 0.0028 | 0.006199 | 0.077251 | 0.060251 | 9.1591 | 0.126001 |

3 | 0.0036 | 0.007970 | 0.087143 | 0.070143 | 8.07762 | 0.139940 |

4 | 0.0044 | 0.009741 | 0.095897 | 0.078897 | 7.30648 | 0.150951 |

5 | 0.0052 | 0.011512 | 0.103813 | 0.086813 | 6.72098 | 0.160820 |

6 | 0.0060 | 0.013283 | 0.111079 | 0.094079 | 6.25689 | 0.169720 |

7 | 0.0068 | 0.015054 | 0.117823 | 0.100823 | 5.87733 | 0.177977 |

8 | 0.0076 | 0.016825 | 0.124134 | 0.107134 | 5.55940 | 0.185710 |

9 | 0.0084 | 0.018596 | 0.130079 | 0.113079 | 5.28804 | 0.193024 |

10 | 0.0092 | 0.020367 | 0.135711 | 0.118711 | 5.05290 | 0.199996 |

11 | 0.0100 | 0.022138 | 0.141069 | 0.124069 | 4.84657 | 0.206677 |

Sr.no . | Q (m^{3}/s)
. | y_{1} (m)
. | y_{2} (m)
. | H (m)
. | Fr1 . | b (m)
. |
---|---|---|---|---|---|---|

1 | 0.0020 | 0.004428 | 0.065680 | 0.048680 | 10.8372 | 0.105097 |

2 | 0.0028 | 0.006199 | 0.077251 | 0.060251 | 9.1591 | 0.126001 |

3 | 0.0036 | 0.007970 | 0.087143 | 0.070143 | 8.07762 | 0.139940 |

4 | 0.0044 | 0.009741 | 0.095897 | 0.078897 | 7.30648 | 0.150951 |

5 | 0.0052 | 0.011512 | 0.103813 | 0.086813 | 6.72098 | 0.160820 |

6 | 0.0060 | 0.013283 | 0.111079 | 0.094079 | 6.25689 | 0.169720 |

7 | 0.0068 | 0.015054 | 0.117823 | 0.100823 | 5.87733 | 0.177977 |

8 | 0.0076 | 0.016825 | 0.124134 | 0.107134 | 5.55940 | 0.185710 |

9 | 0.0084 | 0.018596 | 0.130079 | 0.113079 | 5.28804 | 0.193024 |

10 | 0.0092 | 0.020367 | 0.135711 | 0.118711 | 5.05290 | 0.199996 |

11 | 0.0100 | 0.022138 | 0.141069 | 0.124069 | 4.84657 | 0.206677 |

It is assumed that if the weir behavior is inadequate at *Q*_{min} and *Q*_{max}, it will most likely be problematic at intermediate discharges as well. For both discharges, the leaps have been observed to be relocated downstream. This demonstrates that the cross-sectional flow area of the stepped weir is greater and should be lowered. This will be accomplished by narrowing the step widths because if *‘b*’ varies inversely with *C _{d}*, there is an increase in

*C*.

_{d}## PRACTICAL GEOMETRY OF STEPPED WEIR

Though mathematical procedure renders a stepped weir with 11 steps, practical geometry is adopted for the final section. Details about practical geometry are as follows: The proposed stepped weir in the study has 11 steps. That means 11 risers and 11 treads on either side. Therefore, on either side, there are 22 sharp edges. To avoid the sharp edges and corners in the weir geometry, it is customary to join the steps by a smooth curve. The middle step of width *b*_{1} contributes to almost 80% or greater discharge for any *y*_{2} on its upstream. Except main channel, all remaining steps (2nd–11th) are joined by a smooth curve to obtain the section of ‘CBC designed weir’.

Development of equation for smooth curve joining 10 steps (2nd step–11th step):

The weir is required to be designed as a stepped weir in the initial stage. Then the centers of all treads (from the 2nd step to 11th step) are to be joined by a smooth curved surface.

*Q*

_{max}(10 lps) to

*Q*

_{min}(2 lps) is covered. For existing flume parameters, the width of the channel being 20 cm, 11th step in the proposed mathematical design is discarded. Hence the design will consider 10 steps of rectangular weir in totality. The AutoCAD (version 24.1, 2022) sketch of CBC weir model casted as per theoretical inputs is shown in Figure 4. It has a CBC weir wherein 2nd–10th steps is connected through a creaseless curve.

## EXPERIMENTAL LAYOUT

## NUMERICAL STUDY

The FLOW-3D solver version, created by Flow Science, Inc. in the USA, is the computational fluid dynamics (CFD) model utilized for independent surface stream simulations of flow over the CBC weir. A commercial CFD program called FLOW-3D has special modules created specifically for surface flow applications. FLOW-3D employs the VOF – Volume of the Fluid technique. Applications involving free surface flow are ideally suited for the VOF model (Samadi & Arvanaghi 2014). It is based on the belief that multiple substances should not be mixed together. The grid represents both water and air in this two-phase approach. This method satisfies every water fraction (F) unit, which is unity (1) when the element is totally filled with water and zero (0) whenever it is entirely devoid of water. When the estimate is amid 1 and 0, the component bears an independent water surface.

### Meshing and grid generation

*x*,

*y*, and

*z*axes, a 3D structure was built. The volume of each cube was 5 cm

^{3}, which results in a cell dimension of 1.7 cm. FLOW-3D used a grid of 1, 20,811 cellular components to compute in order to do the baseline CFD analysis with this background. After a 229-second computing period, a steady state condition was identified. Out of the five schemes that make up the FLOW-3D turbulent flow model, currently, the RNG turbulence model is fed in FLOW-3D for CFD, and simulated discharge values are dependent on upstream flow depth. Thereafter, various trials were conducted on CFD for model optimization, indicating the revisions carried out concerning mesh resolution and weir geometry. The meshing and grid generation details are illustrated in Figure 7(a).

### Confinement conditions

The term ‘symmetry’ was used to describe the boundary conditions for side borders, which implies that there is no drag because the same flow takes place on each side of the obstruction. The domain can be effectively halved by employing symmetry boundary conditions, reducing the time required to find a solution. In particular, the symmetry criterion constrains the flow variables. The fluxes are zero across the symmetry. At the same distance from the border, all variables have the same value and gradient. It serves as a mirror, reflecting all flow distribution to the other side. At a symmetric border, there is no flow across the barrier and no scalar flux over the boundary. The *x*-direction barrier had to have ‘defined stagnation pressure.’ By employing this technique, FLOW-3D is able to illustrate a variety of flow heights beginning with a pressure state of stagnation. A continual border output scenario is taken into consideration at the *x*-direction outlet. This illustrated how the flow continued unabated across the border. Figure 7(b) shows the extent state for the *x, y,* and *z* planes.

## RESULTS AND DISCUSSION

The findings from using PVC material cast to operate the laboratory flume at various flow rates are shown (Table 3). As fluctuations in the measure's height and tread occur, conventional rectangular weir equations are utilized to calculate the inequitable release for individual steps that are later empirically estimated for the measured heads. The flume is operated in a variety of ranges (maximum 10 lps to minimum 2 lps) to measure the volumetric discharge. By linking the centrals of subsequent rectangular steps, the weir's curve is created above the base rectangular piece, removing any sharp corners and cavitation risk. The conventional rectangular weir formula is used to calculate the cumulative discharges for various steps.

Sr. no. . | Expt y_{2} (cm)
. | Required time to gather 100 l (s) . | Expt discharge Q (lps)
. | Th. Q (lps) for respective sr. no.
. | C = [Expt _{d}Q/Th. Q] × 0.6
. | CFD Q (lps)
. | C = [CFD Q/Th. Q] × 0.6
. _{d} |
---|---|---|---|---|---|---|---|

1 | 13.57 | 10.96 | 9.118 | 9.2 | 0.594 | 9.24 | 0.602 |

2 | 13.0 | 12.3 | 8.127 | 8.4 | 0.5805 | 7.66 | 0.547 |

3 | 12.413 | 13.65 | 7.322 | 7.6 | 0.578 | 6.95 | 0.548 |

4 | 11.78 | 16.08 | 6.216 | 6.8 | 0.548 | 6.18 | 0.545 |

5 | 11.1 | 18.84 | 5.307 | 6.0 | 0.5307 | 5.42 | 0.542 |

6 | 10.38 | 22.53 | 4.438 | 5.2 | 0.512 | 4.6 | 0.531 |

7 | 9.58 | 27.37 | 3.654 | 4.4 | 0.498 | 4.08 | 0.556 |

8 | 8.714 | 33.52 | 2.983 | 3.6 | 0.497 | 3.44 | 0.573 |

9 | 7.725 | 46.92 | 2.131 | 2.8 | 0.456 | 2.45 | 0.525 |

10 | 6.568 | 54.05 | 1.85 | 2.0 | 0.55 | 1.73 | 0.519 |

Sr. no. . | Expt y_{2} (cm)
. | Required time to gather 100 l (s) . | Expt discharge Q (lps)
. | Th. Q (lps) for respective sr. no.
. | C = [Expt _{d}Q/Th. Q] × 0.6
. | CFD Q (lps)
. | C = [CFD Q/Th. Q] × 0.6
. _{d} |
---|---|---|---|---|---|---|---|

1 | 13.57 | 10.96 | 9.118 | 9.2 | 0.594 | 9.24 | 0.602 |

2 | 13.0 | 12.3 | 8.127 | 8.4 | 0.5805 | 7.66 | 0.547 |

3 | 12.413 | 13.65 | 7.322 | 7.6 | 0.578 | 6.95 | 0.548 |

4 | 11.78 | 16.08 | 6.216 | 6.8 | 0.548 | 6.18 | 0.545 |

5 | 11.1 | 18.84 | 5.307 | 6.0 | 0.5307 | 5.42 | 0.542 |

6 | 10.38 | 22.53 | 4.438 | 5.2 | 0.512 | 4.6 | 0.531 |

7 | 9.58 | 27.37 | 3.654 | 4.4 | 0.498 | 4.08 | 0.556 |

8 | 8.714 | 33.52 | 2.983 | 3.6 | 0.497 | 3.44 | 0.573 |

9 | 7.725 | 46.92 | 2.131 | 2.8 | 0.456 | 2.45 | 0.525 |

10 | 6.568 | 54.05 | 1.85 | 2.0 | 0.55 | 1.73 | 0.519 |

*B*–

*b*). The step width further changes as per every measure's ascent and stride, until (

*y*

_{2}) max is targeted. The proposed weir model's design and discharge value calculations are carried out by the details provided by Hinge

*et al*. (2010). The inequitable release for every measure is produced by the aforementioned calculations as the tread and step height change and is then calculated experimentally for the head observed. Volumetric discharge is calculated by way of jetting the gorge through a variety of ranges (maximum 10 lps to minimum 2 lps). The correction factors are obtained by dividing the actual discharge by the theoretical discharge, and the emitting coefficients are obtained for a sequence of scrutinization (through experimentation) by increasing the ratio of existent theoretical releases by the assumptive

*C*of 0.6 as an input system factor. The head vs discharge behavior reflected from the theoretical approach, experimental model study, and its performance with the CFD approach for CBC weir geometry is shown in Figure 9.

_{d}The current study compares the theoretical model with experimental and CFD performance for broad-crested composite weir design, for which the weir's effectiveness is assessed. The coefficient of discharge values produced by each method are used to illustrate discharge estimation. Theoretical vs experimental vs CFD outputs received for different discharges for CBC weir undertaken here in the study are compared in Table 3. It was discovered that turbulent flow had been produced for the weir flow range, showing that even at larger discharges, the proposed weir had little chance of submerging. *C _{d}* levels were discovered to range from 0.456 to 0.594 during experiments with the CBC model created via additive manufacturing. FLOW-3D was used to explore the CFD pattern, and it yielded relevant results that lined up with the theoretical design.

### Rational estimation model for discharge coefficient

*y*

_{2}) to weir length (

*L*) ratio. The emission acquired for the composite wide-crested weir pattern consisting of various angular transverse segments has been examined in order to develop a suitable predictive regression specimen. Figure 10 demonstrates a clear aligned line fit that has been acquired for all sets of data that exhibit a linear alteration. The coherent sample for for

*C*is outlined by Equation (16) for composite weir cast using additive manufacturing being evaluated in a laboratory tilting flume (with horizontal slope).

_{d}Sr. no. . | 3DP1(C)
. _{d} | CFD 1 . | Theoretical (C)
. _{d} |
---|---|---|---|

1 | 0.594 | 0.602 | 0.6 |

2 | 0.5805 | 0.547 | 0.6 |

3 | 0.578 | 0.548 | 0.6 |

4 | 0.548 | 0.545 | 0.6 |

5 | 0.5307 | 0.542 | 0.6 |

6 | 0.512 | 0.531 | 0.6 |

7 | 0.498 | 0.556 | 0.6 |

8 | 0.497 | 0.573 | 0.6 |

9 | 0.456 | 0.525 | 0.6 |

10 | 0.55 | 0.519 | 0.6 |

C range (Min–Max) _{d} | 0.456–0.594 | 0.519 –0.602 | 0.6 |

Sr. no. . | 3DP1(C)
. _{d} | CFD 1 . | Theoretical (C)
. _{d} |
---|---|---|---|

1 | 0.594 | 0.602 | 0.6 |

2 | 0.5805 | 0.547 | 0.6 |

3 | 0.578 | 0.548 | 0.6 |

4 | 0.548 | 0.545 | 0.6 |

5 | 0.5307 | 0.542 | 0.6 |

6 | 0.512 | 0.531 | 0.6 |

7 | 0.498 | 0.556 | 0.6 |

8 | 0.497 | 0.573 | 0.6 |

9 | 0.456 | 0.525 | 0.6 |

10 | 0.55 | 0.519 | 0.6 |

C range (Min–Max) _{d} | 0.456–0.594 | 0.519 –0.602 | 0.6 |

For estimation of the charted empiric sample, the linear regression method is utilized for deciding *C _{d}* in Equation (16). This rational model has a coefficient of correlation (

*R*) of 0.7135. The model standard error is 0.0183.

When FLOW 3D findings were compared to experimental outputs, both approaches were determined to be accurate under the 3–5% error mentioned in the already published articles (Rady 2011; Bilhan *et al*. 2018). A study performed by Al-Khatib & Gogus (2014), demonstrated discharge models for rectangle compound broad-crested weirs and revealed a range in coefficient of discharge spanning from 0.58 to 1.0 with a mean error of 4.85% between measured and anticipated discharge values. In this context, an investigation of a compound broad-crested weir with a constant discharge coefficient is addressed. The link between theoretical discharges calculated with the classic rectangle weir formulas and observational and CFD discharge for different values is depicted above. The results have a good degree of consistency. The discharge values anticipated by FLOW-3D were a little bit lower than those predicted by the theoretical and practical methods. In contrast to the input design parameter, i.e., 0.6 for CBC weir, the coefficient of discharge must lie between 0.519 and 0.602 as per CFD. Theoretical discharges produced by the conventional linear combination technique were contrasted with the actual and CFD discharges. The CFD analysis predicts discharge values with acceptable accuracy, having mean absolute error (MAE) as −12.82% and *R*^{2} as 0.99. For experimental and CFD research, the discharge coefficient is shown in Table 5. The error sampling for 90, 95, and 99% level of confidence (LOC) having the ‘*p-*value’ for ‘*t*-statistics’ for experimental and CFD *C _{d}* are 0.27 and 1.69%, respectively. The CBC weir model predicts discharges with over 95% confidence, as seen by the

*C*mean value of 0.58 using experimental and numerical approaches in contrast to the 0.6 design input.

_{d}Parameter (C)
. _{d} | Mean (C)
. _{d} | Standard deviation . | t-statistic
. | Confidence level with sampling error . | ||
---|---|---|---|---|---|---|

90% | 95% | 99% | ||||

Expt. | 0.579 | 0.018 | 0.27% | 0.0105 | 0.013 | 0.0187 |

CFD | 0.582 | 0.021 | 1.69% | 0.0126 | 0.0155 | 0.0223 |

Parameter (C)
. _{d} | Mean (C)
. _{d} | Standard deviation . | t-statistic
. | Confidence level with sampling error . | ||
---|---|---|---|---|---|---|

90% | 95% | 99% | ||||

Expt. | 0.579 | 0.018 | 0.27% | 0.0105 | 0.013 | 0.0187 |

CFD | 0.582 | 0.021 | 1.69% | 0.0126 | 0.0155 | 0.0223 |

The key contribution of this research work is that it investigates composite broad-crested weirs using a novel approach in which the discharge coefficient is the primary input design parameter. This study presents the research as a possibility to forge the creative notion of keeping steady *C _{d}* regardless of the head over weir, which has been highlighted through numerous tests undertaken and described herein. Another benefit of the proposed composite hydraulic structure is the inclusion of a low crest height in the design model, which will enable sediments to pass through high and low discharges while maintaining the reliability of the discharge calibration curve. Lower weir heights in permanent watercourses will enable continuous fish flow, preserving ecological balance. Artificial fish ladders are not required. In addition, the actual and numerical results achieved when compared to the theoretical design, the picture is positive. These findings would be beneficial for practising field engineers, especially when a general discharge equation and composite weir discharge coefficient are unavailable. A researcher can determine discharge corresponding to head over CBC weir by using the head discharge rating curve, regardless of weir shape or the coefficient of discharge for different flow ranges. The research objectives undertaken behind the study are seen accomplished.

## CONCLUSION

With discussions and inferences drawn regarding CBC weir model investigations carried out, it can be stated that the proposed weir behaves efficiently keeping the discharge coefficient constant hereby fulfilling the novelty of the proposed research work. The continuity observed in the rating curves justifies that the proposed composite weir with broad crest measures discharges for different upstream head variations. A CBC weir of multiple rectangular steps with different step widths is designed and calibrated with an experimental approach. The link between head–discharge is established for composite with a broad-crested weir combination of rectangular and parabolic weirs by theoretical, experimental, and numerical approaches to compute a wide range of discharge. The mould of the composite weir sample with the help of an addible constructing application is seen to have a variance in *C _{d}* that ranges from 0.45 to 0.594 in the experimental output. The mean interval estimate of 0.58 value of the suggested CBC weir structure is close to 95% accuracy while estimating the discharge coefficient. A make-and-cut investigation of the stepwise treatment of weir shape will definitely yield a constant

*C*value which is a novel finding in this research. The investigations and inferences about composite wide-mouthed weir reported here, are believed to be the first of their kind to maintain a constant coefficient of discharge for many different discharges, to the best of knowledge. Hence there are no practical examples of such kind of CBC weir with rectangular and parabolic shapes so far. The present study is limited to CBC weir applications designed for rectangular prismatic channel sections. Its workability in the case of other prismatic channel sections like trapezoidal and triangular shapes could be explored.

_{d}## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*A new technique for controlling the location of hydraulic jump in a rectangular channel*

*Open Channel Flow Measurement Handbook*(1.1)